Problem: In three-dimensional space, find the number of lattice points that have a distance of 3 from the origin.

Note: A point is a lattice point if all its coordinates are integers.
Solution: Let the point be $(x,y,z).$  Each coordinate can only be 0, $\pm 1,$ $\pm 2,$ or $\pm 3.$  Checking we find that up to sign, the only possible combinations of $x,$ $y,$ and $z$ that work are either two 0s and one 3, or one 1 and two 2s.

If there are two 0s and one 3, then there are 3 ways to place the 3.  Then the 3 can be positive or negative, which gives us $3 \cdot 2 = 6$ points.

If there is one 1 and two 2s, then there are 3 ways to place the 1.  Then each coordinate can be positive or negative, which gives us $3 \cdot 2^3 = 24$ points.

Therefore, there are $6 + 24 = \boxed{30}$ such lattice points.